Merkurjev--Suslin Theorem

The Merkurjev--Suslin theorem (1982) was the first major case of the Bloch--Kato conjecture, establishing the norm residue isomorphism in degree 2. It provides a complete description of the $n$ -torsion in the Brauer group of a field in terms of Milnor K-theory and has deep applications to central simple algebras and quadratic forms.

Statement

Theorem4.3Merkurjev--Suslin Theorem

For any field $F$ and any integer $n \geq 1$ with $\operatorname{char}(F) \nmid n$ , the norm residue homomorphism

$h_n^2: K_2^M(F)/n \xrightarrow{\;\sim\;} H^2_{\text{et}}(F, \mu_n^{\otimes 2})$

is an isomorphism. When $F$ contains a primitive $n$ -th root of unity, $\mu_n^{\otimes 2} \cong \mu_n$ and

$K_2^M(F)/n \cong H^2(G_F, \mu_n) \cong \operatorname{Br}(F)[n]$

where $\operatorname{Br}(F)[n]$ is the $n$ -torsion in the Brauer group.

Interpretation for the Brauer group

RemarkCyclic algebras generate the Brauer group

The Merkurjev--Suslin theorem has the following concrete interpretation. For a field $F$ containing $\mu_n$ :

Every central simple $F$ -algebra $A$ of exponent dividing $n$ (i.e., $[A]^n = 1$ in $\operatorname{Br}(F)$ ) can be written as a tensor product of cyclic algebras:

$A \sim (a_1, b_1)_n \otimes_F (a_2, b_2)_n \otimes_F \cdots \otimes_F (a_r, b_r)_n$

in the Brauer group, where $(a, b)_n$ denotes the cyclic algebra generated by $u, v$ with $u^n = a$ , $v^n = b$ , $vu = \zeta_n uv$ .

Before Merkurjev--Suslin, it was unknown whether every element of $\operatorname{Br}(F)$ is a sum of symbols. The theorem shows that symbols generate, though the minimal number of symbols needed (the symbol length) can be arbitrarily large.

ExampleThe case n = 2: quaternion algebras

For $n = 2$ (Merkurjev's earlier theorem, 1981): every 2-torsion element of $\operatorname{Br}(F)$ is a sum of quaternion algebras. That is, every central simple algebra of exponent 2 is Brauer equivalent to a tensor product of quaternion algebras:

$A \sim (a_1, b_1) \otimes \cdots \otimes (a_r, b_r)$

where $(a, b) = F \oplus Fi \oplus Fj \oplus Fij$ with $i^2 = a$ , $j^2 = b$ , $ij = -ji$ .

This implies Pfister's conjecture: the class of the Clifford algebra $C(q)$ in $\operatorname{Br}(F)$ (for a quadratic form $q$ ) lies in the image of $K_2^M(F)/2 \to \operatorname{Br}(F)[2]$ . Since Pfister forms generate $I^2/I^3$ in the Witt ring, this connects to the Milnor conjecture on quadratic forms.

Key elements of the proof

Proof

The proof of the Merkurjev--Suslin theorem is long and technical. We outline the main ideas.

Step 1: Surjectivity (easier direction). Show $h_n^2$ is surjective, i.e., every $n$ -torsion Brauer class is a sum of cyclic algebras.

This uses the Severi--Brauer variety: for $[A] \in \operatorname{Br}(F)$ of degree $n$ , the Severi--Brauer variety $\operatorname{SB}(A)$ is a smooth projective variety of dimension $n-1$ with $\operatorname{SB}(A)(F) \neq \emptyset \iff A$ is split. Over the generic point $F(\operatorname{SB}(A))$ , the algebra splits, giving a class in $K_0(\operatorname{SB}(A))$ .

Using the localization sequence and the K-theory of Severi--Brauer varieties:

$K_0(\operatorname{SB}(A)) \cong \mathbb{Z}^n / (\text{relations from } A)$

one shows that the obstruction to splitting lies in $K_2^M(F)/n$ , establishing surjectivity.

Step 2: Injectivity (harder direction). Show that if $\sum \{a_i, b_i\} \in K_2^M(F)$ maps to zero in $\operatorname{Br}(F)[n]$ , then $\sum \{a_i, b_i\} = 0$ in $K_2^M(F)/n$ .

This proceeds by induction on the "complexity" of the symbol. The key tool is the norm principle: if a symbol $\{a, b\}$ splits over a finite extension $L/F$ (i.e., is zero in $K_2^M(L)/n$ ), then

$\{a, b\} = N_{L/F}(\text{something in } K_2^M(L)/n)$

where $N_{L/F}$ is the norm map.

Step 3: Transfer and specialization. For the induction, one specializes to function fields of Severi--Brauer varieties and uses the exact sequence:

$K_2^M(F) / n \to K_2^M(F(X)) / n \to \bigoplus_{x \in X^{(1)}} K_1^M(k(x)) / n$

where $X = \operatorname{SB}(A)$ . The boundary maps and norm maps interact to provide the inductive step.

Step 4: Conclusion. Combining surjectivity and injectivity, $h_n^2$ is an isomorphism. $\square$

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Applications

ExampleApplications to algebra and geometry

Index-exponent problem: For a CSA $A/F$ , the exponent $e = \operatorname{exp}(A)$ (order in $\operatorname{Br}(F)$ ) divides the index $d = \operatorname{ind}(A) = \sqrt{\dim_F D}$ where $D$ is the division algebra Brauer-equivalent to $A$ . The Merkurjev--Suslin theorem gives bounds on the index in terms of the exponent for specific classes of fields.
$u$ -invariant: For the field $F = \mathbb{Q}_p(\!(t)\!)$ , the Merkurjev--Suslin theorem combined with local class field theory shows that every quadratic form of dimension $> 8$ over $F$ is isotropic ( $u(F) = 8$ ).
Essential dimension: The theorem implies $\operatorname{ed}(PGL_n) \geq 2$ for $n \geq 2$ (the essential dimension of the projective linear group). This was later improved using higher Bloch--Kato.
Suslin's rigidity: As a byproduct, Suslin proved that for an algebraically closed field $k$ , the map $K_n(k) \otimes \mathbb{Z}/m \to K_n^{\text{top}}(\mathbb{C}) \otimes \mathbb{Z}/m$ is an isomorphism, establishing a form of rigidity for K-theory with finite coefficients.